Primorial

With primorial (from English primorial), or prime faculty , one describes the product of all prime numbers that do not exceed a certain number. The terms are closely related to the faculty and are mainly used in the mathematical field of number theory .

The name Primorial is the Germanized English word primorial . The product of the prime numbers less than or equal to , however, is seldom called the primorial in German , and even more rarely the prime faculty . It is usually paraphrased as the “ product of the prime numbers less than or equal to n ”. ${\displaystyle n}$

definition

For a natural number , the prime faculty is defined as the product of all prime numbers less than or equal to : ${\displaystyle n}$ ${\displaystyle n\#}$ ${\displaystyle n}$

${\displaystyle n\#=\prod _{p\ \mathrm {prim} }^{p\leq n}p}$.

Sometimes a distinction is made between the special case in which there is a prime number and the primorial is defined analogously only for this , which remains undefined for non-prime . ${\displaystyle n}$ ${\displaystyle n}$

In this case , the product is empty, the value of the prime faculty and the primorial is then 1. The primorial has no values for arguments that are not prime numbers. The prime faculty delivers the value for this that the next lower prime number would deliver. In practical use, however, both terms are mostly used as synonyms. ${\displaystyle n\leq 1}$ ${\displaystyle n}$ ${\displaystyle n}$

example

In order to calculate the value of the primorial , one first determines all prime numbers less than or equal to 7. These are 2, 3, 5 and 7. The product of these four prime numbers yields . For 9, on the other hand, one could not calculate a primorial, but the prime faculty - since 9 is not a prime number and the next lower prime number is 7 and the next higher prime number is 11, the following applies . ${\displaystyle 7\#}$ ${\displaystyle 7\#=2\cdot 3\cdot 5\cdot 7=210}$ ${\displaystyle 7\#=8\#=9\#=10\#=210}$

properties

Comparison of the faculty (yellow) and the prime faculty (red)

• Let and be two neighboring prime numbers. Then for every natural number with : ${\displaystyle p}$ ${\displaystyle q}$ ${\displaystyle n}$ ${\displaystyle p\leq n<q}$

${\displaystyle n\#=p\#}$

• The following estimate is known for the primorial

${\displaystyle n\#\leq 4^{n}}$.

• The following also applies:

${\displaystyle \lim _{n\to \infty }{\sqrt[{n}]{n\#}}=e}$

The values for are smaller than , but with larger values, the values ​​of the function exceed the limit and later oscillate around infinitely often . ${\displaystyle n<10^{11}}$ ${\displaystyle e}$ ${\displaystyle n}$ ${\displaystyle e}$ ${\displaystyle e}$

• If the -th is prime, then has exactly divisors. For example, the number has 2 factors, has 4 factors, 8 and already has factors, because 97 is the 25th prime number. ${\displaystyle p_{k}}$ ${\displaystyle k}$ ${\displaystyle p_{k}\#}$ ${\displaystyle 2^{k}}$ ${\displaystyle 2\#}$ ${\displaystyle 3\#}$ ${\displaystyle 5\#}$ ${\displaystyle 97\#}$ ${\displaystyle 2^{25}}$

• The sum of the reciprocal values ​​of the prime faculty converges to a constant

${\displaystyle \sum _{p{\text{ prim}}}{1 \over p\#}={1 \over 2}+{1 \over 6}+{1 \over 30}+\ldots =0{,}7052301717918\ldots }$

The Engel development (a special stem fraction development ) of this number forms the sequence of the prime numbers (see sequence A064648 in OEIS )

• According to Euclid's theorem , is used to prove the infinity of all prime numbers. ${\displaystyle p\#+1}$

Table with example values

n	n #
2	2
3	6th
5	30th
7th	210
11	2,310
13	30,030
17th	510.510
19th	9,699,690
23	223.092.870
29	6,469,693,230
31	200.560.490.130
37	7.420.738.134.810
41	304.250.263.527.210
43	13,082,761,331,670,030
47	614,889,782,588,491,410
53	32,589,158,477,190,044,730
59	1,922,760,350,154,212,639,070
61	117,288,381,359,406,970,983,270
67	7,858,321,551,080,267,055,879,090
71	557,940,830,126,698,960,967,415,390
73	40.729.680.599.249.024.150.621.323.470
79	3,217,644,767,340,672,907,899,084,554,130
83	267,064,515,689,275,851,355,624,017,992,790
89	23,768,741,896,345,550,770,650,537,601,358,310
97	2,305,567,963,945,518,424,753,102,147,331,756,070

swell

1. ^ GH Hardy, EM Wright: An Introduction to the Theory of Numbers . 4th edition. Oxford University Press, Oxford 1975. ISBN 0-19-853310-1 .
   Theorem 415, p. 341
2. ^ L. Schoenfeld: Sharper bounds for the Chebyshev functions and ${\displaystyle \theta (x)}$ ${\displaystyle \psi (x)}$ . II. Math. Comp. Vol. 34, No. 134 (1976) 337-360; there p. 359.
   Quoted in: G. Robin: Estimation de la fonction de Tchebychef sur le -ieme nombre premier et grandes valeurs de la fonction , nombre de diviseurs premiers de ${\displaystyle \theta }$ ${\displaystyle k}$ ${\displaystyle \omega (n)}$ ${\displaystyle n}$ . Acta Arithm. XLII (1983) 367-389 ( PDF 731KB ); there p. 371

Web links

• Primary Faculty and Primorial